Impossible intersections in a Weierstrass family of elliptic curves

TL;DR

This paper characterizes parameters in a Weierstrass family of elliptic curves where two points are simultaneously torsion, showing such parameters are rare or nonexistent under certain conditions, and improves results on torsion points in Legendre families.

Contribution

It provides explicit criteria for when two points are simultaneously torsion in a Weierstrass family, extending previous work and refining conditions for the absence of such parameters.

Findings

Set of parameters with simultaneous torsion points is empty unless ratio is -2 or -1/2.

Set remains empty under certain 2-adic conditions even when ratio is not rational.

Improves upon recent results on torsion points in Legendre elliptic curves.

Abstract

Consider the Weierstrass family of elliptic curves $E_{\lambda}:y^2=x^3+\lambda$ parametrized by nonzero $\lambda\in\overline{\mathbb{Q}_2}$, and let $P_{\lambda}(x)=(x,\sqrt{x^3+\lambda})\in E_{\lambda}$. In this article, given $\alpha,\beta\in\overline{\mathbb{Q}_2}$ such that $\frac{\alpha}{\beta}\in\mathbb{Q}$, we provide an explicit description for the set of parameters $\lambda$ such that $P_{\lambda}(\alpha)$ and $P_{\lambda}(\beta)$ are simultaneously torsion for $E_{\lambda}$. In particular we prove that the aforementioned set is empty unless $\frac{\alpha}{\beta}\in\{-2,-\frac{1}{2}\}$. Furthermore, we show that this set is empty even when $\frac{\alpha}{\beta}\notin\mathbb{Q}$ provided that $\alpha$ and $\beta$ have distinct $2-$adic absolute values and the ramification index $e(\mathbb{Q}_2(\frac{\alpha}{\beta})~\vert~\mathbb{Q}_2)$ is coprime with $6$. We also improve upon a recent result of Stoll concerning the Legendre family of elliptic curves $E_{\lambda}:y^2=x(x-1)(x-\lambda)$, which itself strengthened earlier work of Masser and Zannier by establishing that provided $a,b$ have distinct reduction modulo $2$, the set $\{\lambda\in\mathbb{C}\setminus\{0,1\}~:~(a,\sqrt{a(a-1)(a-\lambda)}),(b,\sqrt{b(b-1)(b-\lambda)})\in (E_{\lambda})_{tors}\}$ is empty.